Step response worksheet with answers

Free printable Step response worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
t (s)y(t)1000.5 s60

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 100. At t = 0.5 s it reads 60. Find the time constant τ.

y(∞) = 100y(0.5) = 60
τ = s
2.Easy
t (s)y(t)12.50.8 s?

A first-order system G(s) = 2.5/(0.8s + 1) receives a step input of size 5 at t = 0, starting from rest. Find the output y(0.8) at t = 0.8 s.

G(s) = 2.5/(0.8s + 1)U = 5t = 0.8 s
y(0.8) =
3.Medium
t (s)y(t)3tp​ = ?

A system has the transfer function G(s) = 291/(s² + 4s + 97) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 291/(s² + 4s + 97)
tp = s
4.Medium
t (s)y(t)2

A system has the transfer function G(s) = 28/(s² + 5s + 14) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 28/(s² + 5s + 14)
σ =
5.Hard
k = 100 N/mc = 13 N·s/mm = 5 kgF = 10 Nx(t)

A mass m = 5 kg hangs on a spring k = 100 N/m with a viscous damper c = 13 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 10 N is suddenly applied. Find the damping ratio ζ.

m = 5 kgk = 100 N/mc = 13 N·s/mF = 10 N
ζ =
6.Hard
R(s)+Ks(s + 5)G(s)C(s)−

A unity negative feedback loop has G(s) = K/(s(s + 5)). Find the gain K that gives a step response with 25 % overshoot.

G(s) = K/(s(s + 5))%OS = 25 %
K =

Answer key

  1. 1. τ = 0.5457 s
  2. 2. y(0.8) = 7.902
  3. 3. tp = 0.3258 s
  4. 4. σ = −2.5
  5. 5. ζ = 0.2907
  6. 6. K = 38.35

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